Question

Heya ! can anyone help me with this ?

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Answer 1

See below

Explanation:

`\text{if } (\sin ^4 x)/(2) + (\cos^4 x)/(3) = (1)/(5), \text{ then show that } (\sin ^8 x)/(8) + (\cos^8 x)/(27)=(1)/(125)`

Considering the identity

`\boxed{ \sin ^2 x+\cos^2 x = 1}`

then, we have
`\sin ^2 x+\cos^2 x = 1 \iff \cos^2 x = 1 -\sin ^2 x`, thus

`(\sin ^4 x)/(2) + ((\cos^2 x)^2)/(3) =(\sin ^4 x)/(2) + (( 1 -\sin ^2)^2)/(3)`

`\text{For } \sin^2x =y: (y^2)/(2) + (( 1 -y)^2)/(3) = ( 3y^2)/(3\cdot2) + (2( 1 -y)^2)/(2\cdot3) = (3y^2 + 2( 1 -y)^2)/(6)`

`= (3y^2 + 2-4y+2y^2)/(6) = (5y^2 -4y+2)/(6)`

We are considering

`(5y^2 -4y+2)/(6) = (1)/(5) \iff (5y^2 -4y+2)/(6) - (1)/(5) = 0 \iff (25y^2-20y+4)/(30) =0`

Therefore, multiplying both sides by 30,

`25y^2-20y+4 = 0`

Using the quadratic equation,

`y=(-(-20)\pm √((-20)^2-4\cdot 25\cdot 4))/(2\cdot 25) = (20\pm √(0))/(50) = (20)/(50) = (2)/(5)`

The roots are equal.

Once

`y=(2)/(5) \implies \sin^2 x = (2)/(5)`

Again, considering the identity

`\sin ^2 x+\cos^2 x = 1`

`\sin^2 x = (2)/(5) \implies (2)/(5)+\cos^2 x = 1 \iff \cos^2 x = (3)/(5)`

Substituting both values in the initial equations,

`((4)/(25) )/(2) + ((9)/(25))/(3) = (1)/(5) \implies (4)/(50) +(9)/(75) = (1)/(5) \implies (2)/(25) +(3)/(25) = (1)/(5)\implies (1)/(5) = (1)/(5)`

`((16)/(625))/(8) + ((81)/(625))/(27)=(1)/(125)\implies (16)/(5000) + (81)/(16875) = (1)/(125) \implies (2)/(625)+(3)/(625) = (1)/(125)`

`\implies (1)/(125)= (1)/(125)`

`\therefore (\sin ^4 x)/(2) + (\cos^4 x)/(3) = (1)/(5) \iff (\sin ^8 x)/(8) + (\cos^8 x)/(27)=(1)/(125)`